In the last few posts we looked at how government spending and taxation interact with the circular flow of money to increase aggregate incomes. This process was predicated on an injection of money that is gradually depleted as it is taxed out of circulation during repeated spending. In the interim period an amount of aggregate income is generated which is greater than the injected amount.

Just for the sake of it, we'll revisit a simpler case, without government money, and see if we can understand the circulation of money in terms of a similar multiplier description and accounting framework.

When we first looked at the circular flow of money, we envisaged a fixed amount of money already in existence simply being passed around. We used the simple equations:

$$Y = C$$$$C = Y$$

The first equation simply describes the identity which pairs spending with incomes - every act of spending generates an equivalent amount of income for someone else. The second equation is a statement about the behaviour of folk in our economy. It states that folk simply spend all of their income: spending is equal to income. We call this type of equation a consumption function.

This is certainly a highly simplified model of an economy with ridiculously implausible assumptions and ludicrous ommissions. but even such daft models can have plenty of heuristic value. In this case, there are some issues that we can improve on and that might help to nail down some of the concepts that might be useful in more complicated models.

Money stock/wealth not reference Only reference to flows suggests infinite speed have to specify or infer velocity, but this always implies that folk are holding money at some stage Where did money start?

had to know how many times the total money stock was spent in a particular time period in order to know the total cumulative spending and income. This was described in terms of the velocity of money. Then when we initially looked at the fiscal multiplier, we found that

$$H_t = H_0$$$$Y_t = C_t$$$$C_t = \alpha_Y Y_t + \alpha_H H_{t-1}$$

...

$$Y_t = \alpha_Y Y_t + \alpha_H H_{t-1}$$$$Y_t (1 - \alpha_Y) = \alpha_H H_{t-1}$$$$Y_t = \frac {\alpha_H H_{t-1}}{1 - \alpha_Y}$$$$Y_t = \frac {\alpha_H }{1 - \alpha_Y} H_{t-1}$$$$Y_t = \frac {\alpha_H }{1 - \alpha_Y} H_{0}$$



In [6]:

    
alpha_h/(1-alpha_y)*100









    Out[6]:





50.000000000000014



In [19]:

    
import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline  

N = 200

alpha_y = 0.9*1.0
alpha_h = 0.5

C   = np.zeros(N) # consumption
Y   = np.zeros(N) # income
H_h = np.zeros(N) # private wealth

C[0] = 0
Y[0] = 0
H_h[0] = 100

for t in range(1, N):
    
    # calculate consumer spending
    C[t] = alpha_y*Y[t-1] + alpha_h*H_h[t-1] 
        
    # calculate total income (consumer spending plus constant government spending)
    Y[t]   = alpha_h/(1-alpha_y)*H_h[t-1] 
    
    H_h[t] = H_h[t-1] +(1-alpha_y)*Y[t] - alpha_h*H_h[t-1]



In [20]:

    
fig = plt.figure(figsize=(12, 8))

consumption_plot = fig.add_subplot(131, xlim=(0, N), ylim=(0, np.max(Y)))
consumption_plot.plot(range(N), C, lw=3)
consumption_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('consumption')

income_plot = fig.add_subplot(132, xlim=(0, N), ylim=(0, np.max(Y)))
income_plot.plot(range(N), Y, lw=3)
income_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('income')


debt_plot = fig.add_subplot(133, xlim=(0, N), ylim=(0,np.max(H_h)*1.5))
debt_plot.plot(range(N), H_h, lw=3)
debt_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government debt')

# space subplots neatly
plt.tight_layout()



In [ ]: